Causal Autoregressive Time Series

In the MAT8181graduate course on Time Series, we will discuss (almost) only causal models. For instance, with ,

with some white noise , those models are obtained when . In that case, we’ve seen that was actually the innovation process, and we can write

which is actually a mean-square convergent series (using simple Analysis arguments on series). From that expression, we can easily see that is stationary, since (which does not depend on ) and

(which does not depend on ).

Consider now the case where . Clearly, we have some problem here, since

cannot be defined (the series does not converge, in ). Nevertheless, it is still possible to write

But it is possible to iterate (as in the previous case) and write

which is actually well defined. And in that case, the sequence of random variables obtained from this equation is the unique stationary solution of the recursive equation . This might be confusing, but the thing is this solution should not be confused with the usual non-stationary solution of obtained from . As in the code writen to generate a time series, from some starting value in the previous post.

Now, let us spent some time with this stationary time series, considered as unatural in Brockwell and Davis (1991). One point is that, in the previous case (where ) was the innovation process. So variable was not correlated with the future of the noise, . Which is not the case when .

All that looks nice, if you’re willing to understand thing at some theoretical level. What does all that mean from a computational perspective ? Consider some white noise (this noise actually does exist whatever you want to define, based on that time series)